OFFSET
0,3
COMMENTS
The denominators are given in A300299.
The sum given in the name is computed using a telescopic sum. See the general recipe given in the Jolley reference, (201), p. 38.
REFERENCES
L. B. W. Jolley, Summation of Series, Dover Publications, 2nd rev. ed., 1961, p. 38, (201).
FORMULA
a(n) = numerator(r(n)), with the result of the sum given in the name r(n) = n*(50 + 35*n + 10*n^2 + n^3)/(96*(1 + n)*(2 + n)*(n + 3)*(4 + n)), n >= 0.
This results from r(n) = 1/96 - 1/(4*(1+n)*(2+n)*(n+3)*(4+n)).
G.f. for rationals {r(n)}_{n >= 0}: (1/96)*(1 - hypergeometric([1, 4], [5], -x/(1-x)))/(1-x)
= (-x*(12 - 42*x + 52*x^2 - 25*x^3) + 12*(1 - x)^4*log(1/(1-x))) / (288*x^4*(1-x)).
EXAMPLE
The sum begins: 0 + 1/(1*2*3*4*5) + 1/(2*3*4*5*6) + ... = 0 + 1/120 + 1/720 + 1/2520 + 1/6720 + 1/15120 + 1/30240 + ...
The rationals r(n) (partial sums) begin: 0/1, 1/120, 7/720, 17/1680, 23/2240, 125/12096, 209/20160, 329/31680, 247/23760, 119/11440, 125/12012, 341/32760, ...
MATHEMATICA
Table[Numerator[n (50 + 35 n + 10 n^2 + n^3) / (96 (1 + n) (2 + n) (n + 3) (4 + n))], {n, 0, 50}] (* Vincenzo Librandi, Apr 06 2018 *)
PROG
(GAP) List(List([0..40], n->Sum([0..n-1], k->1/(Product([0..4], j->k+j+1)))), NumeratorRat); # Muniru A Asiru, Apr 05 2018
(PARI) a(n) = numerator(sum(k=0, n-1, prod(j=0, 4, (k+j+1))^(-1))); \\ Altug Alkan, Apr 05 2018
(Magma) [Numerator(n*(50+35*n+10*n^2+n^3)/(96*(1+n)*(2+n)*(n+3)*(4+n))): n in [0..50]]; // Vincenzo Librandi, Apr 06 2018
CROSSREFS
KEYWORD
nonn,frac,easy
AUTHOR
Wolfdieter Lang, Apr 05 2018
STATUS
approved